In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of $\text{FP}_n$-injective objects to be a torsion class. By doing so, propose notion $n$-hereditary categories. We also define study $\text{FP}_n$-flat in categories with generating set small projective objects, provide several equivalent this torsion-free. end, present applications examples contex...

Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R ! R is called a generalized derivation on R if there exists a derivation d : R to R such that F(xy) = F(x)y + xd(y) holds for all x y in R. In the present paper we describe the action of generalized derivations satisfying several conditions on Lie ideals of semiprime rings.

Background Abnormal myocardial rotation and torsion have been investigated earlier as markers of ventricular failure1-2. However, there are no previous studies on global or regional torsion in functional single ventricle (SV) or the contribution of torsion to biomechanical differences between single RV (SRV) and single LV (SLV) hearts. Additionally, regional variations in rotation between the v...

It is known that the pure braid groups are residually torsion-free nilpotent. This property is however widely open for the most obvious generalizations of these groups, like pure Artin groups and like fundamental groups of hyperplane complements (even reflection ones). In this paper we relate this problem to the faithfulness of linear representations, and prove the residual torsion-free nilpote...

We prove injectivity and relative asphericity for "layered" systems of equations over torsion-free groups, when the exponent matrix is invertible over Z.

Let H be a torsion-free δ-hyperbolic group with respect to a finite generating set S. Let g1, . . . , gn and g1∗, . . . , gn∗ be elements of H such that gr∗ is conjugate to gr for each r = 1, . . . , n. There is a uniform conjugator if and only if W (g1∗, . . . , gn∗) is conjugate to W (g1, . . . , gn) for every word W in n variables and length up to a computable constant depending only on δ, ♯...

The Atiyah conjecture for a discrete group G states that the L-Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of G. Here we establish conditions under which the Atiyah conjecture for a torsion-free group G implies the Atiyah conjecture for every fi...

cyclic groups, torsion-free extensions, Betti numbers We describe algorithms to determine extensions of infinite polycyclic groups having certain properties. In particular, we are interested in torsion-free extensions and extensions whose Fitting subgroup has a minimal centre. Then we apply these methods in topological applications. We discuss the calculation of Betti numbers for compact manifo...

We study the question when a ∗-autonomous (Mix-)category has a representation as a ∗-autonomous category of a compact one. We prove that necessary and sufficient condition is that weak distributivity maps are monic (or, equivalently epic). For a Mix-category, this condition is, in turn, equivalent to the requirement that Mix-maps be monic (or epic). We call categories satisfying this property t...